D₅₃×#2194 #27538019

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

D₅₃×#2194

Splittings

Fitting	C₅₃×#2194 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2194

Subgroups

	subgroup	G/H
center	C₂	#447×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2194	C₂
socle	C₂×C₅₃	C₂×#447
prime radical	#576×D₅₃	C₂
hypercenter	#2194	D₅₃
hypocenter	C₅₃	C₂×#2194
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #2194

1130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 4 generators on 85 points:

(1,3)(2,8)(4,13)(5,15)(6,16)(7,9)(10,27)(11,28)(12,22)(14,19)(17,25)(18,20)(21,23)(24,32)(26,30)(29,31)(33,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35);(1,3)(2,7)(4,18)(5,14)(6,16)(8,9)(10,24)(11,26)(12,17)(13,20)(15,19)(21,31)(22,25)(23,29)(27,32)(28,30);(1,2,6,11)(3,12,16,27)(4,17,21,32)(5,22,19,10)(7,23,26,13)(8,15,28,14)(9,29,30,20)(18,24,31,25);(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)

Conjugacy classes & character table

644 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323495);

D53×#2194 #27538019